A structure-preserving approach to simulating Hamiltonian systems with dissipation

  • Jeyakumar, Sandra (Australian National University)
  • Kraus, Michael (Max-Planck-Institut für Plasmaphysik)
  • Pfefferlé, David (University of Western Australia)
  • Hole, Matthew (Australian National University)

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Hamiltonian systems occur naturally in many physical problems, including those in plasma physics. The behaviour of charged particles in electromagnetic fields, which consistently influence each other, can be modelled through the Vlasov-Maxwell equations — a 6D time-dependent system of Hamiltonian hyperbolic partial differential equations. This Hamiltonian structure has recently been exploited in the design of numerical solvers, as it provides one way to ensure the numerical preservation of invariants (or first integrals) associated with the problem. Unfortunately, introducing dissipation into such systems (most often through global integral operators) can spoil this pure Hamiltonian treatment. In this talk, we present the metriplectic approach [1], which is a formalism for producing structure-preserving methods for Hamiltonian systems with dissipation. This works to reproduce the dissipative terms through using an entropy-like quantity and metric bracket that are constructed to be compatible with the Hamiltonian structure, in essence ensuring compatibility with the first law of thermodynamics. This structure can then be carried into the numerical discretisation of the problem as well. The talk will present an introduction to this topic, illustrated through relevant examples and numerical results. [1] Morrison, P. J. (1986). A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena, 18(1–3), 410–419,